经典位势论及其对应的概率论 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

经典位势论及其对应的概率论 英文PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Part 1 Classical and Parabolic Potential Theory3

Chapter Ⅰ Introduction to the Mathematical Background of Classical Potential Theory3

1．The Context of Green's Identity3

2．Function Averages4

3．Harmonic Functions4

4．Maximum-Minimum Theorem for Harmonic Functions5

5．The Fundamental Kernel for RN and Its Potentials6

6．Gauss Integral Theorem7

7．The Smoothness of Potentials;The Poisson Equation8

8．Harmonic Measure and the Riesz Decomposition11

Chapter Ⅱ Basic Properties of Harmonic,Subharmonic,and Superharmonic Functions14

1．The Green Function of a Ball;The Poisson Integral14

2．Harnack's Inequality16

3．Convergence of Directed Sets of Harmonic Functions17

4．Harmonic,Subharmonic,and Superharmonic Functions18

5．Minimum Theorem for Superharmonic Functions20

6．Application of the Operation τB20

7．Characterization of Superharmonic Functions in Terms of Harmonic Functions22

8．Differentiable Superharmonic Functions23

9．Application of Jensen's Inequality23

10．Superharmonic Functions on an Annulus24

11．Examples25

12．The Kelvin Transformation(N≥2)26

13．Greenian Sets27

14 The L1(μB-)and D(μB-)Classes of Harmonic Functions on a Ball B;The Riesz-Herglotz Theorem27

15．The Fatou Boundary Limit Theorem31

16．Minimal Harmonic Functions33

Chapter Ⅲ Infima of Families of Superharmonic Functions35

1．Least Superharmonic Majorant(LM) and Greatest Subharmonic Minorant(GM)35

2．Generalization of Theorem 136

3．Fundamental Convergence Theorem(Preliminary Version)37

4．The Reduction Operation38

5．Reduction Properties41

6．A Smallness Property of Reductions on Compact Sets42

7．The Natural(Pointwise)Order Decomposition for Positive Superharmonic Functions43

Chapter Ⅳ Potentials on Special Open Sets45

1．Special Open Sets,and Potentials on Them45

2．Examples47

3．A Fundamental Smallness Property of Potentials48

4．Increasing Sequences of Potentials49

5．Smoothing of a Potential49

6．Uniqueness of the Measure Determining a Potential50

7．Riesz Measure Associated with a Superharmonic Function51

8．Riesz Decomposition Theorem52

9．Counterpart for Superharmonic Functions on R2 of the Riesz Decomposition53

10 An Approximation Theorem55

Chapter Ⅴ Polar Sets and Their Applications57

1．Definition57

2．Superharmonic Functions Associated with a Polar Set58

3．Countable Unions of Polar Sets59

4．Properties of Polar Sets59

5．Extension of a Superharmonic Function60

6．Greenian Sets in R2 as the Complements of Nonpolar Sets63

7．Superharmonic Function Minimum Theorem(Extension of Theorem II.5)63

8．Evans-Vasilesco Theorem64

9．Approximation of a Potential by Continuous Potentials66

10．The Domination Principle67

11．The Infinity Set of a Potential and the Riesz Measure68

Chapter Ⅵ The Fundamental Convergence Theorem and the Reduction Operation70

1．The Fundamental Convergence Theorem70

2．Inner Polar versus Polar Sets71

3．Properties of the Reduction Operation74

4．Proofs of the Reduction Properties77

5．Reductions and Capacities84

Chapter Ⅶ Green Functions85

1．Definition of the Green Function GD85

2．Extremal Property of GD87

3．Boundedness Properties of GD88

4．Further Properties of GD90

5．The Potential GDμ of a Measure μ92

6．Increasing Sequences of Open Sets and the Corresponding Green Function Sequences94

7．The Existence of GD versus the Greenian Character of D94

8．From Special to Greenian Sets95

9．Approximation Lemma95

10．The Function GD(·,ζ)|D-{ζ} as a Minimal Harmonic Function96

Chapter Ⅷ The Dirichlet Problem for Relative Harmonic Functions98

1．Relative Harmonic,Superharmonic,and Subharmonic Functions98

2．The PWB Method99

3．Examples104

4．Continuous Boundary Functions on the Euclidean Boundary(h≡1)106

5．h-Harmonic Measure Null Sets108

6．Properties of PWBh Solutions110

7．Proofs for Section 6111

8．h-Harmonic Measure114

9．h-Resolutive Boundaries118

10．Relations between Reductions and Dirichlet Solutions122

11．Generalization of the Operator τh B and Application to GMh123

12．Barriers124

13．h-Barriers and Boundary Point h-Regularity126

14．Barriers and Euclidean Boundary Point Regularity127

15．The Geometrical Significance of Regularity(Euclidean Boundary,h≡1)128

16．Continuation of Section 13130

17．h-Harmonic Measure μh D as a Function ofD131

18．The Extension G＝ D of GD and the Harmonic Average μD(ξ,G＝ B(η,·))When D ？ B132

19．Modification of Section 18 for D=R2136

20．Interpretation of φD as a Green Function with Pole ∞(N=2)139

21．Variant of the Operator τB140

Chapter Ⅸ Lattices and Related Classes of Functions141

1．Introduction141

2．LMh D u for an h-Subharmonic Function u141

3．The Class D(μh D-)142

4．The Class Lp(μh D-)(p≥1)144

5．The Lattices(S±,≤)and(S+,≤)145

6．The Vector Lattice (S,≤)146

7．The Vector Lattice Sm148

8．The Vector Lattice Sp149

9．The Vector Lattice Sqb150

10．The Vector Lattice Ss151

11．A Refinement ofthe Riesz Decomposition152

12．Lattices of h-Harmonic Functions on a Ball152

Chapter Ⅹ The Sweeping Operation155

1．Sweeping Context and Terminology155

2． Relation between Harmonic Measure and the Sweeping Kernel157

3．Sweeping Symmetry Theorem158

4．Kernel Property of δA D158

5．Swept Measures and Functions160

6．Some Properties of δA D161

7．Poles of a Positive Harmonic Function163

8．Relative Harmonic Measure on a Polar Set164

Chapter Ⅺ The Fine Topology166

1．Definitions and Basic Properties166

2．A Thinness Criterion168

3．Conditions That ξ∈A∫169

4．An Internal Limit Theorem171

5．Extension of the Fine Topology to RN∪{∞}175

6．The Fine Topology Derived Set of a Subset of RN177

7．Application to the Fundamental Convergence Theorem and to Reductions177

8．Fine Topology Limits and Euclidean Topology Limits178

9．Fine Topology Limits and Euclidean Topology Limits(Continued)179

10．Identification of A∫ in Terms ofa Special Functionu180

11．Quasi-Lindel？f Property180

12．Regularity in Terms of the Fine Topology181

13．The Euclidean Boundary Set of Thinness of a Greenian Set182

14．The Support of a Swept Measure183

15．Characterization of ‖μ‖A183

16．A Special Reduction184

17．The Fine Interior of a Set of Constancy of a Superharmonic Function184

18．The Support of a Swept Measure (Continuation of Section 14)185

19．Superharmonic Functions on Fine-Open Sets187

20．A Generalized Reduction187

21．Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains190

22．The Limit Harmonic Measure ∫μD191

23．Extension of the Domination Principle194

Chapter Ⅻ The Martin Boundary195

1．Motivation195

2．The Martin Functions196

3．The Martin Space197

4．Preliminary Representations of Positive Harmonic Functions and Their Reductions199

5．Minimal Harmonic Functions and Their Poles200

6．Extension of Lemma 4201

7．The Set of Nonminimal Martin Boundary Points202

8．Reductions on the Set of Minimal Martin Boundary Points203

9．The Martin Representation204

10．Resolutivity of the Martin Boundary207

11．Minimal Thinness at a Martin Boundary Point208

12．The Minimal-Fine Topology210

13．First Martin Boundary Counterpart of Theorem XI.4(c)and(d)213

14．Second Martin Boundary Counterpart of Theorem XI.4(c)213

15．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point215

16．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point(Continued)216

17．Minimal-Fine Martin Boundary Limit Functions216

18．The Fine Boundary Function of a Potential218

19．The Fatou Boundary Limit Theorem for the Martin Space219

20．Classical versus Minimal-Fine Topology Boundary Limit Theorems for Relative Superharmonic Functions on a Ball in RN221

21．Nontangential and Minimal-Fine Limits at a Half-space Boundary222

22．Normal Boundary Limits for a Half-space223

23．Boundary Limit Function (Minimal-Fine and Normal)of a Potential on a Half-space225

Chapter ⅩⅢ Classical Energy and Capacity226

1．Physical Context226

2．Measures and Their Energies227

3．Charges and Their Energies228

4．Inequalities between Potentials,and the Corresponding Energy Inequalities229

5．The Function D？GDμ230

6．Classical Evaluation of Energy;Hilbert Space Methods231

7．The Energy Functional(Relative to an Arbitrary Greenian Subset D of RN)233

8．Alternative Proofs ofTheorem 7(b+)235

9．Sharpening of Lemma 4237

10．The Classical Capacity Function237

11．Inner and Outer Capacities(Notation of Section 10)240

12．Extremal Property Characterizations of Equilibrium Potentials(Notation of Section 10)241

13．Expressions for C(A)243

14．The Gauss Minimum Problems and Their Relation to Reductions244

15．Dependence of C on D247

16．Energy Relative to R2248

17．The Wiener Thinness Criterion249

18．The Robin Constant and Equilibrium Measures Relative to R2(N=2)251

Chapter ⅩⅣ One-Dimensional Potential Theory256

1．Introduction256

2．Harmonic,Superharmonic,and Subharmonic Functions256

3．Convergence Theorems256

4．Smoothness Properties of Superharmonic and Subharmonic Functions257

5．The Dirichlet Problem(Euclidean Boundary)257

6．Green Functions258

7．Potentials of Measures259

8．Identification of the Measure Defining a Potential259

9．Riesz Decomposition260

10．The Martin Boundary261

Chapter ⅩⅤ Parabolic Potential Theory:Basic Facts262

1．Conventions262

2．The Parabolic and Coparabolic Operators263

3．Coparabolic Polynomials264

4．The Parabolic Green Function of RN266

5．Maximum-Minimum Parabolic Function Theorem267

6．Application of Green's Theorem269

7．The Parabolic Green Function of a Smooth Domain;The Riesz Decomposition and Parabolic Measure(Formal Treatment)270

8．The Green Function of an Interval272

9．Parabolic Measure for an Interval273

10．Parabolic Averages275

11．Harnack's Theorems in the Parabolic Context276

12．Superparabolic Functions277

13．Superparabolic Function Minimum Theorem279

14．The Operation ？ and the Defining Average Properties of Superparabolic Functions280

15．Superparabolic and Parabolic Functions on a Cylinder281

16．The Appell Transformation282

17．Extensions of a Parabolic Function Defined on a Cylinder283

Chapter ⅩⅥ Subparabolic,Superparabolic,and Parabolic Functions on a Slab285

1．The Parabolic Poisson Integral fora Slab285

2．A Generalized Superparabolic Function Inequality287

3．A Crrterion of a Subparabolic Function Supremum288

4．A Boundary Limit Criterion for the Identically Vanishing of a Positive Parabolic Function288

5．A Condition that a Positive Parabolic Function Be Representable by a Poisson Integral290

6．The L1(？-)and D(？-)Classes of Parabolic Functions on a Slab290

7．The Parabolic Boundary Limit Theorem292

8．Minimal Parabolic Functions on a Slab293

Chapter ⅩⅦ Parabolic Potential Theory (Continued)295

1．Greatest Minorants and Least Majorants295

2．The Parabolic Fundamental Convergence Theorem(Preliminary Version)and the Reduction Operation295

3．The Parabolic Context Reduction Operations296

4．The Parabolic Green Function298

5．Potentials300

6．The Smoothness of Potentials303

7．Riesz Decomposition Theorem305

8．Parabolic-Polar Sets305

9．The Parabolic-Fine Topology308

10．Semipolar Sets309

11．Preliminary List of Reduction Properties310

12．A Criterion of Parabolic Thinness313

13．The Parabolic Fundamental Convergence Theorem314

14．Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions316

15．Applications of the Fundamental Convergence Theorem to the Parabolic-Fine Topology317

16．Parabolic-Reduction Properties317

17．Proofs of the Reduction Properties in Section 16320

18．The Classical Context Green Function in Terms of the Parabolic Context Green Function(N≥1)326

19．The Quasi-Lindel？f Property328

Chapter ⅩⅧ The Parabolic Dirichlet Problem,Sweeping,and Exceptional Sets329

1．Relativization of the Parabolic Context;The PWB Method in this Context329

2．h-Parabolic Measure332

3．Parabolic Barriers333

4．Relations between the Classical Dirichlet Problem and the Parabolic Context Diriehlet Problem334

5．Classical Reductions in the Parabolic Context335

6．Parabolic Regularity of Boundary Points337

7．Parabolic Regularity in Terms of the Fine Topology341

8．Sweeping in the Parabolic Context341

9．The Extension ？of ？ and the Parabolic Average ？(？,？(·,？)when ？343

10．Conditions that ξ∈？ps345

11．Parabolic-and Coparabolic-Polar Sets347

12．Parabolic-and Coparabolic-Semipolar Sets348

13．The Support of a Swept Measure350

14．An Internal Limit Theorem;The Coparabolic-Fine Topology Smoothness of Superparabolic Functions351

15．Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab357

16．The Parabolic Context Domination Principle358

17．Limits of Superparabolic Functions at Parabolic-Irregular Boundary Points of Their Domains358

18．Martin Flat Point Set Pairs361

19．Lattices and Related Classes of Functions in the Parabolic Context361

Chapter ⅩⅨ The Martin Boundary in the Parabolic Context363

1．Introduction363

2．The Martin Functions of Martin Point Set and Measure Set Pairs364

3．The Martin Space ？M366

4．Preparatory Material for the Parabolic Context Martin Representation Theorem367

5．Minimal Parabolic Functions and Their Poles369

6．The Set of Nonminimal Martin Boundary Points370

7．The Martin Representation in the Parabolic Context371

8 Martin Boundary of a Slab ？=RN×]0,δ[with 0＜δ≤+∞371

9．Martin Boundaries for the Lower Half-space of？Nand for ？N374

10．The Martin Boundary of？=]0,+∞[×]-∞,δ[375

11．？WB？Solutions on ？M377

12．The Minimal-Fine Topology in the Parabolic Context377

13．Boundary Counterpart ofTheorem ⅩⅧ.14(f)379

14．The Vanishing ofPotentials on ？M？381

15．The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces381

Part 2 Probabilistic Counterpart of Part1387

Chapter Ⅰ Fundamental Concepts of Probability387

1．Adapted Families of Functions on Measurable Spaces387

2．Progressive Measurability388

3．Random Variables390

4．Conditional Expectations391

5．Conditional Expectation Continuity Theorem393

6．Fatou's Lemma for Conditional Expectations396

7．Dominated Convergence Theorem for Conditional Expectations397

8．Stochastic Processes,“Evanescent,""Indistinguishable,""Standard Modification,""Nearly"398

9．The Hitting of Sets and Progressive Measurability401

10．Canonical Processes and Finite-Dimensional Distributions402

11．Choice of the Basic Probability Space404

12．The Hitting of Sets by a Right Continuous Process405

13．Measurability versus Progressive Measurability of Stochastic Processes407

14．Predictable Families of Functions410

Chapter Ⅱ Optional Times and Associated Concepts413

1．The Context of Optional Times413

2．Optional Time Properties(Continuous Parameter Context)415

3．Process Functions at Optional Times417

4．Hitting and Entry Times419

5．Application to Continuity Properties of Sample Functions421

6．Continuation of Section 5423

7．Predictable Optional Times423

8．Section Theorems425

9．The Graph of a Predictable Time and the Entry Time of a Predictable Set426

10．Semipolar Subsets of R+×Ω427

11．The Classes D and Lp of Stochastic Processes428

12．Decomposition of Optional Times;Accessible and Totally Inaccessible Optional Times429

Chapter Ⅲ Elements of Martingale Theory432

1．Definitions432

2．Examples433

3．Elementary Properties(Arbitrary Simply Ordered Parameter Set)435

4．The Parameter Set in Martingale Theory437

5．Convergence of Supermartingale Families437

6．Optional Sampling Theorem(Bounded Optional Times)438

7．Optional Sampling Theorem for Right Closed Processes440

8．Optional Stopping442

9．Maximal Inequalities442

10．Conditional Maximal Inequalities444

11．An Lp Inequality for Submartingale Suprema444

12．Crossings445

13．Forward Convergence in the L1 Bounded Case450

14．Convergence ofa Uniformly Integrable Martingale451

15．Forward Convergence of a Right Closable Supermartingale453

16．Backward Convergence of a Martingale454

17．Backward Convergence of a Supermartingale455

18．The τ Operator455

19．The Natural Order Decomposition Theorem for Supermartingales457

20．The Operators LM and GM458

21．Supermartingale Potentials and the Riesz Decomposition459

22．Potential Theory Reductions in a Discrete Parameter Probability Context459

23．Application to the Crossing Inequalities461

Chapter Ⅳ Basic Properties of Continuous Parameter Supermartingales463

1．Continuity Properties463

2．Optional Sampling of Uniformly Integrable Continuous Parameter Martingales468

3．Optional Sampling and Convergence of Continuous Parameter Supermartingales470

4．Increasing Sequences of Supermartingales473

5．Probability Version of the Fundamental Convergence Theorem of Potential Theory476

6．Quasi-Bounded Positive Supermartingales;Generation of Supermartingale Potentials by Increasing Processes480

7．Natural versus Predictable Increasing Processes(I=Z+ or R+)483

8．Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case488

9．An Inequality for Predictable Increasing Processes489

10．Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets490

11．Generation of Supermartingale Potentials by Increasing Processes in the Continuous Parameter Case:The Meyer Decomposition493

12．Meyer Decomposition of a Submartingale495

13．Role of the Measure Associated with a Supermartingale;The Supermartingale Domination Principle496

14．The Operators τ,LM,and GM in the Continuous Parameter Context500

15．Potential Theory on R+×Ω501

16．The FineTopology of R+× Ω502

17．Potential Theory Reductions in a Continuous Parameter Probability Context504

18．Reduction Properties505

19．Proofs ofthe Reduction Properties in Section 18509

20．Evaluation of Reductions513

21．The Energy of a Supermartingale Potential515

22．The Subtraction of a Supermartingale Discontinuity516

23．Supermartingale Decompositions and Discontinuities518

Chapter Ⅴ Lattices and Related Classes of Stochastic Processes520

1．Conventions;The Essential Order520

2．LMx(·)when{x(·),F(·)}Is a Submartingale521

3．Uniformly Integrable Positive Submartingales523

4．LpBounded Stochastic Processes(p≥1)524

5．The Lattices('S±,≤),('S+,≤),(S±,≤),(S+,≤)525

6．The Vector Lattices('S,≤)and(S,≤)528

7．The Vector Lattices('Sm,≤)and(Sm,≤)529

8．The Vector Lattices('Sp,≤)and(Sp,≤)530

9．The Vector Lattices('Sqb,≤)and(Sqb,≤)531

10．The Vector Lattices('Ss,≤)and(Ss,≤)532

11．The Orthogonal Decompositions'Sm='Smqb+'Sms and Sm=Smqb+Sms533

12．Local Martingales and Singular Supermartingale Potentials in(S,≤)534

13．Quasimartingales(Continuous Parameter Context)535

Chapter Ⅵ Markov Processes539

1．The Markov Property539

2．Choice of Filtration544

3．Integral Parameter Markov Processes with Stationary Transition Probabilities545

4．Application of Martingale Theory to Discrete Parameter Markov Processes547

5．Continuous Parameter Markov Processes with Stationary Transition Probabilities550

6．Specialization to Right Continuous Processcs552

7．Continuous Parameter Markov Processes:Lifctimes and Trap Points554

8．Right Continuity of Markov Process Filtrations;A Zero-One(0-1)Law556

9．Strong Markov Property557

10．Probabilistic Potential Theory;Excessive Functions560

11．Excessive Functions and Supermartingales564

12．Excessive Functions and the Hitting Times of Analytic Sets(Notation and Hypotheses of Section 11)565

13．Conditioned Markov Processes566

14．Tied Down Markov Processes567

15．Killed Markov Processes568

Chapter Ⅵ Brownian Motion570

1．Processes with Independent Increments and State Space RN570

2．Brownian Motion572

3．Continuity of Brownian Paths576

4．Brownian Motion Filtrations578

5．Elementary Properties of the Brownian Transition Density and Brownian Motion581

6．The Zero-One Law for Brownian Motion583

7．Tied Down Brownian Motion586

8．André Reflection Principle587

9．Brownian Motion in an Open Set(N≥1)589

10．Space-Time Brownian Motion in an Open Set592

11．Brownian Motion in an Intervai594

12．Probabilistic Evaluation of Parabolic Measure for an Interval595

13．Probabilistic Significance of the Heat Equation and Its Dual596

Chapter Ⅷ599

The It？ Integral599

1．Notation599

2．The Size of Г0601

3．Properties ofthe It？ Integral602

4．The Stochastic Integral for an Integrand Process in Г0605

5．The Stochastic Integral for an Integrand Process in Г606

6．Proofs of the Properties in Section 3607

7．Extension to Vector-Valued and Complex-Valued Integrands611

8．Martingales Relative to Brownian Motion Filtrations612

9．A Change of Variables615

10．The Role of Brownian Motion Increments618

11．(N=1)Computation of the It？ Integral by Riemann-Stieltjes Sums620

12．It？'s Lemma621

13．The Composition of the Basic Functions of Potential Theory with Brownian Motion625

14．The Composition of an Analytic Function with Brownian Motion626

Chapter Ⅸ Brownian Motion and Martingale Theory627

1．Elementary Martingale Applications627

2．Coparabolic Polynomials and Martingale Theory630

3．Superharmonic and Harmonic Functions on RN and Supermartingales and Martingales632

4．Hitting of an Fσ Set635

5．The Hitting of a Set by Brownian Motion636

6．Superharmonic Functions,Excessive for Brownian Motion637

7．Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion;A Probabilistic Fatou Boundary Limit Theorem641

8．Excessive and Invariant Functions for Brownian Motion645

9．Application to Hitting Probabilities and to Parabolicity of Transition Densities647

10．(N=2).The Hitting of Nonpolar Sets by Brownian Motion648

11．Continuity of the Composition of a Function with Brownian Motion649

12．Continuity of Superharmonic Functions on Brownian Motion650

13．Preliminary Probabilistic Solution of the Classical Dirichlet Problem651

14．Probabilistic Evaluation of Reductions653

15．Probabilistic Description of the Fine Topology656

16．α-Excessive Functions for Brownian Motion and Their Composition with Brownian Motions659

17．Brownian Motion Transition Functions as Green Functions;The Corresponding Backward and Forward Parabolic Equations661

18．Excessive Measures for Brownian Motion663

19．Nearly Borel Sets for Brownian Motion666

20．Brownian Motion into a Set from an Irregular Boundary Point666

Chapter Ⅹ Conditional Brownian Motion668

1．Definition668

2．h-Brownian Motion in Terms of Brownian Motion671

3．Contexts for(2.1)676

4．Asymptotic Character of h-Brownian Paths at Their Lifetimes677

5．h-Brownian Motion from an Infinity of h680

6．Brownian Motion under Time Reversal682

7．Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic Functions;h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions684

8．Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions688

9．Conditional Brownian Motion in a Ball691

10．Conditional Brownian Motion Last Hitting Distributions;The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution693

11．The TailσAlgebra ofa Conditional Brownian Motion694

12．Conditional Space-Time Brownian Motion699

13．[Space-Time]Brownian Motion in[？N]RN with Parameter Set R700

Part3705

Chapter Ⅰ Lattices in Classical Potential Theory and Martingale Theory705

1．Correspondence between Classical Potential Theory and Martingale Theory705

2．Relations between Decomposition Components of S in Potential Theory and Martingale Theory706

3．The Classes Lp and D706

4．PWB-Related Conditions on h-Harmonic Functions and on Martingales707

5．Class D Property versus Quasi-Boundedness708

6．A Condition for Quasi-Boundedness709

7．Singularity of an Element of S+ m710

8．The Singular Component of an Element of S+711

9．The Class Spqb712

10．The Class Sps714

11．Lattice Theoretic Analysis of the Composition of an h-Superharmonic Function with an h-Brownian Motion715

12．A Decomposition of S+ ms(Potential Theory Context)716

13．Continuation of Section 11717

Chapter Ⅱ Brownian Motion and the PWB Method719

1．Context of the Problem719

2．Probabilistic Analysis of the PWB Method720

3．PWBh Examples723

4．Tail σ Algebras in the PWBh Context725

Chapter Ⅲ Brownian Motion on the Martin Space727

1．The Structure of Brownian Motion on the Martin Space727

2．Brownian Motions from Martin Boundary Points(Notation of Section 1)728

3．The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology(Notation of Section 1)730

4．The Probabilistic Fatou Theorem on the Martin Space732

5．Probabilistic Approach to Theorem 1.XI.4(c)and Its Boundary Counterparts733

6．Martin Representation of Harmonic Functions in the Parabolic Context735

Appendixes741

Appendix Ⅰ Analytic Sets741

1．Pavings and Algebras of Sets741

2．Suslin Schemes741

3．Sets Analytic over a Product Paving742

4．Analytic Extensions versus σ Algebra Extensions of Pavings743

5．Projection Characterization A(y)743

6．The Operation A(A)744

7．Projections of Sets in Product Pavings744

8．Extension of a Measurability Concept to the Analytic Operation Context745

9．The Gδ Sets of a Complete Metric Space745

10．Polish Spaces746

11．The Baire Null Space746

12．Analytic Sets747

13．Analytic Subsets of Polish Spaces748

Appendix Ⅱ Capacity Theory750

1．Choquet Capacities750

2．Sierpinski Lemma750

3．Choquet Capacity Theorem751

4．Lusin's Theorem751

5．A Fundamental Example of a Choquet Capacity752

6．Strongly Subadditive Set Functions752

7．Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function753

8．Topological Precapacities755

9．Universally Measurable Sets756

Appendix Ⅲ Lattice Theory758

1．Introduction758

2．Lattice Definitions758

3．Cones758

4．The Specific Order Generated by a Cone759

5．Vector Lattices760

6．Decomposition Property of a Vector Lattice762

7．Orthogonality in a Vector Lattice762

8．Bands in a Vector Lattice762

9．Projections on Bands763

10．The Orthogonal Complement of a Set764

11．The Band Generated by a Single Element764

12．Order Convergence765

13．Order Convergence on a Linearly Ordered Set766

Appendix Ⅳ Lattice Theoretic Concepts in Measure Theory767

1．Lattices of Set Algebras767

2．Measurable Spaces and Measurable Functions767

3．Composition of Functions768

4．The Measure Lattice of a Measurable Space769

5．The σ Finite Measure Lattice of a Measurable Space(Notation of Section 4)771

6．The Hahn and Jordan Decompositions772

7．The Vector Lattice Mσ772

8．Absolute Continuity and Singularity773

9．Lattices of Measurable Functions on a Measure Space774

10．Order Convergence of Families of Measurable Functions775

11．Measures on Polish Spaces777

12．Derivates of Measures778

Appendix Ⅴ Uniform Integrability779

Appendix Ⅵ Kernels and Transition Functions781

1．Kernels781

2．Universally Measurable Extension of a Kernel782

3．Transition Functions782

Appendix Ⅶ Integral Limit Theorerns785

1．An Elementary Limit Theorem785

2．Ratio Integral Limit Theorems786

3．A One-Dimensional Ratio Integral Limit Theorem786

4．A Ratio Integral Limit Theorem Involving Convex Variational Derivates788

Appendix Ⅷ Lower Semicontinuous Functions791

1．The Lower Sernicontinuous Smoothing of a Function791

2．Suprema of Families of Lower Semicontinuous Functions791

3．Choquet Topological Lemma792

Historical Notes793

Part 1793

Part 2806

Part 3815

Appendixes816

Bibliography819

Notation Index827

Index829